In this paper, we consider the class of discrete singular hyperbolic dynamical systems. These are hyperbolic maps \(f : K \setminus N \rightarrow K\) , where \(K \subset M\)  is a precompact open subset of a Riemannian manifold \(M\) , and \(N \subset K\)  is a closed subset of singularities on which \(f\)  fails to be continuous and/or differentiable. The map \(f\)  is uniformly hyperbolic on the non-invariant set \(K \setminus N\) , but behaves more similarly to the non-uniformly hyperbolic setting on an invariant set that consists of trajectories passing nearby the the singular set \(N\)  with a prescribed rate. Our setting includes systems that are derived from Poincaré maps of hyperbolic flows, such as the geometric Lorenz attractor, but also includes singular hyperbolic dynamical systems that do not arise from flows, such as the Lozi map [1]}. In [2]}, it was shown that the attractors admitted by singular hyperbolic maps support at most countably many ergodic SRB measures. In [3]}, conditions were given under which a singular hyperbolic attractor admits at most finitely many ergodic SRB measures. We provide an alternative proof of this result, with somewhat different conditions that are easy to verify. Namely, if the singular set is a disjoint union of finitely many embedded submanifolds that transversally intersect unstable leaves, and if the image of neighborhoods of the singular set remain separated from the singular set under the dynamics for sufficiently long but finite time (conditions (SH3), (SH6), and (SH7)), then there are at most finitely SRB measures.
